Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

We show that the L^p spatial–temporal decay rates of solutions of incompressible flow in an 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in He, Xin [C. He, Z. Xin, Weighted estimates for nonstationary Navier–Stokes equations in exterior domain, Methods Appl. Anal. 7 (3) (2000) 443–458], and our previous results [H.-O. Bae, B.J. Jin, Asymptotic behavior of Stokes solutions in 2D exterior domains, J. Math. Fluid Mech., in press; H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, submitted for publication]. For the spatial decay rate estimate, we first extend temporal decay rate result of the Navier–Stokes solutions for general L^p space when the initial velocity is in , 1<rq<∞ (1<r<q=∞).     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Large time behavior for the non‐isentropic Navier–Stokes–Maxwell system

In this paper, we are concerned with the system of the non‐isentropic compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time‐decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large‐time behavior is based on the linearized analysis of the non‐isentropic Navier–Stokes–Poisson equations and the electromagnetic part for the linearized isentropic Navier–Stokes–Maxwell equations. In the meantime, the time‐decay rates obtained by Zhang, Li, and Zhu [J. Differential Equations, 250(2011), 866‐891] for the linearized non‐isentropic Navier–Stokes–Poisson equations are improved. Copyright © 2016 John Wiley & Sons, Ltd.     

Local exact controllability of the Navier–Stokes system

In this paper we deal with the local exact controllability of the Navier–Stokes system with distributed controls supported in small sets. In a first step, we present a new Carleman inequality for the linearized Navier–Stokes system, which leads to null controllability at any time T>0. Then, we deduce a local result concerning the exact controllability to the trajectories of the Navier–Stokes system.     

Blow-up of viscous heat-conducting compressible flows

We show the blow-up of strong solution of viscous heat-conducting flow when the initial density is compactly supported. This is an extension of Z. Xin's result [Z. Xin, Blow up of smooth solutions to the compressible Navier–Stokes equations with compact density, Comm. Pure Appl. Math. 51 (1998) 229–240] to the case of positive heat conduction coefficient but we do not need any information for the time decay of total pressure nor the lower bound of the entropy. We control the lower bound of second moment by total energy and obtain the exact relationship between the size of support of initial density and the existence time. We also provide a sufficient condition for the blow-up in case that the initial density is positive but has a decay at infinity.     

A jump discontinuity of compressible viscous flows grazing a non-convex corner

We show existence and regularity of solution for the compressible viscous steady state Navier–Stokes system on a polygon having a grazing corner and that the density has a jump discontinuity across a curve inside the domain. There are corresponding jumps in derivatives of the velocity. The solution comes from a well-posed boundary value problem on a polygonal domain with a non-convex corner. A formula for the decay of the jump is given. The decay formula suggests that density jumps can occur in a compressible flow with a non-vanishing viscosity.     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution

In this paper our objective is to provide physically reasonable solutions for the stationary Navier–Stokes equations in a two-dimensional domain with two outlets to infinity, a semi-strip Π₋ and a half-plane K. The same problem in an aperture domain, i.e. in a domain with two half-plane outlets to infinity, has been studied but only under symmetry restrictions on the data. Here, we assume that the main asymptotic term of the solution takes an antisymmetric form in K and apply the technique of weighted spaces with detached asymptotics, i.e. we use spaces where the functions have prescribed asymptotic forms in the outlets.After first showing that the corresponding Stokes problem admits a unique solution if and only if certain compatibility conditions are satisfied, we write the Navier–Stokes equations as a perturbation of the Stokes problem and the crucial compatibility condition as an algebraic equation by which the flux becomes determined. Assuming that the coefficient of the main (antisymmetric) asymptotic term of the solution in K does not vanish and that the data are sufficiently small, we use a contraction principle to solve the Navier–Stokes system coupled with the algebraic equation.Finally, we discuss the ill-posedness of the Navier–Stokes problem with prescribed flux.     

Two-level Galerkin–Lagrange multipliers method for the stationary Navier–Stokes equations

In this paper, we combine the Galerkin–Lagrange multiplier (GLM) method with the two-level method to solve the stationary Navier–Stokes equations in order to avoid the time-consuming process and the construction of zero-divergence elements. Different quadrilateral partitions are used for approximating the velocity and the pressure. Then some error estimates are obtained and some numerical results of the GLM method and the two-level GLM method are given. The results show that the two-level method based on the GLM method is more efficient than the GLM method under the convergence rate of same order.     

Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations

We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H^{3/2+ε,3/4+ε/2}(Q), with 0<ε. In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t=0, is achieved by choosing a time varying control operator in a neighbourhood of t=0.     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.     

On Navier–Stokes Equations with Slip Boundary Conditions in an Infinite Pipe

The paper examines steady Navier–Stokes equations in a two-dimensional infinite pipe with slip boundary conditions. At both inlet and outlet, the velocity of flow is assumed to be constant. The main results show the existence of weak and regular solutions with no restrictions of smallness of the flux vector, also simply connectedness of the domain is not required.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

On the global existence and time decay estimates in critical spaces for the Navier–Stokes–Poisson system

We are concerned with the study of the Cauchy problem for the Navier–Stokes–Poisson system in the critical regularity framework. In the case of a repulsive potential, we first establish the unique global solvability in any dimension for small perturbations of a linearly stable constant state. Next, under a suitable additional condition involving only the low frequencies of the data and in the L²‐critical framework (for simplicity), we exhibit optimal decay estimates for the constructed global solutions, which are similar to those of the barotropic compressible Navier–Stokes system. Our results rely on new a priori estimates for the linearized Navier–Stokes–Poisson system about a stable constant equilibrium, and on a refined time‐weighted energy functional.     

On the existence of solutions to the Navier–Stokes–Poisson equations of a two‐dimensional compressible flow

In this paper, we consider the Navier–Stokes–Poisson equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(?)=a?log^d (?) for large ?. Here d>1 and a>0. Copyright © 2006 John Wiley & Sons, Ltd.     

Structure of the set of stationary solutions of phase-lock equations in superconductivity

In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164].     

A note on integration of the Navier–Stokes equations

In this study the 2D Navier–Stokes equations are used to obtain a new self-similar equation. The latter equation, subject to appropriate boundary conditions and volume discharge, describes the pressure distribution and velocity field of a plane free jet.     

Navier–Stokes equations with nonhomogeneous boundary conditions in a convex bi-dimensional domain

We prove the existence of a weak solution to Navier–Stokes equations describing the isentropic flow of a gas in a convex and bounded region, ΩR², with nonhomogeneous Dirichlet boundary conditions on ∂Ω. These results are also extended to flow domain surrounding an obstacle.     

A Flux Correction Multigrid for Compressible Flow

A finite-volume based linear multigrid algorithm is proposed and used within an implicit linearized scheme to solve Navier–Stokes equations for compressible laminar flows. Coarse level problems are constructed algebraically based on convective and diffusive fluxes, without the knowledge of coarse geometry. Numerical results for complex 2D geometries such as airfoils, including stretched meshes, show mesh size independent convergence and efficiency of the method compared to other finite-volume-based multigrid method.     

Zero‐viscosity‐capillarity limit to rarefaction waves for the 1D compressible Navier–Stokes–Korteweg equations

In this paper, we study the zero viscosity and capillarity limit problem for the one‐dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd.